Module 2 Quiz: The Derivative
Quiz Instructions
Answer each question, then check. Aim for 7+ out of 10.
1. Find f'(x) for f(x) = 6x5 − 2x3 + 4.
30x4 − 6x²
2. What is the instantaneous rate of change of g(x) = x² + 3x at x = 2?
g'(x) = 2x + 3. g'(2) = 7.
3. Find dy/dx for y = 4√x − 2/x.
y = 4x1/2 − 2x−1. dy/dx = 2x−1/2 + 2x−2 = 2/√x + 2/x².
4. Find the tangent line to f(x) = x4 at x = 1.
f(1)=1, f'(x)=4x³, f'(1)=4. y = 4x − 3.
5. Where is f(x) = |x − 2| not differentiable?
At x = 2 (corner).
6. Find d/dx [3ex + 5 ln x].
3ex + 5/x.
7. True or false: Differentiable at x = a implies continuous at x = a.
True. Differentiability always implies continuity.
8. Find f''(x) for f(x) = x³ − 6x² + x.
f' = 3x² − 12x + 1. f'' = 6x − 12.
9. A ball's height is h(t) = −16t² + 96t. Find its velocity at t = 3.
v(t) = −32t + 96. v(3) = −96+96 = 0 (at its peak).
10. Find all x where the tangent to y = x³ − 12x has slope 0.
y' = 3x² − 12 = 0. x² = 4. x = ±2.